$3$-Cocycle invariants

Let $X$ be a finite quandle and $A$ an abelian group.

A function $\theta : X \times X \times X \rightarrow A$ is called a quandle $3$-cocycle if it satisfies the $3$-cocycle condition

$${\theta (x,z,w)- \theta(x,y,w)+\theta(x,y,z) -\theta(x*y,z,w)}$$

$$+\theta(x*z,y*z,w)-\theta(x*w,y*w,z*w)=0, \quad \forall x,y,z, w \in X,$$

and $\theta(x,x,y)=0=\theta(x,y,y), \forall x, y \in X$.

Let ${\cal C}$ be a coloring of arcs and regions of a given diagram $K$. Let $(x,y,z) (=(x_{\tau}, y_{\tau}, z_{\tau}))$ be the ordered triple of colors at a crossing $\tau$, see Fig. . Let $\theta$ be a $3$-cocycle. Then the weight in this case is defined by $B( {\cal C}, \tau)=\phi(x_{\tau}, y_{\tau}, z_{\tau})^{\epsilon(\tau)}$ where $\epsilon(\tau)$ is $\pm 1$ for positive and negative crossing, respectively. Then the ($3$-)cocycle invariant is defined in a similar way to $2$-cocycle invariants by $\Phi_{\phi}(K) =\{ \sum_{\tau} B( {\cal C}, \tau ) \vert {\cal C}\in {\rm Col_X(K)} \}$. The multiset version is defined similarly.